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biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic. Formally, a biholomorphic function is a function...

between charts are biholomorphic, complex manifolds are, in particular, smooth and canonically oriented (not just orientable: a biholomorphic map to (a subset...

around every point on the sphere there is a neighborhood that can be biholomorphically identified with C {\displaystyle \mathbf {C} } . On the other hand...

is not all of C {\displaystyle \mathbb {C} } , then there exists a biholomorphic mapping f {\displaystyle f} (i.e. a bijective holomorphic mapping whose...

Thus, under this definition, a map is conformal if and only if it is biholomorphic. The two definitions for conformal maps are not equivalent. Being one-to-one...

American Mathematical Society. In 1977 he proved a significant theorem on biholomorphic mappings between algebraic real hypersurfaces. Using his expertise on...

Riemann surfaces M {\displaystyle M} and N {\displaystyle N} are called biholomorphic (or conformally equivalent to emphasize the conformal point of view)...

n>1} , open balls and open polydiscs are not biholomorphically equivalent, that is, there is no biholomorphic mapping between the two. This was proven by...

domain is a proper subdomain of C n {\displaystyle \mathbb {C} ^{n}} , biholomorphically equivalent to C n {\displaystyle \mathbb {C} ^{n}} . That is, an open...

Kähler manifold has positive bisectional curvature, then it must be biholomorphic to complex projective space. In this way, it can be viewed as an analogue...

complex plane. The Riemann mapping theorem implies that there is a biholomorphic mapping f from the upper half-plane { ζ ∈ C : Im ⁡ ζ > 0 } {\displaystyle...

isometry group. In the category of Riemann surfaces, an automorphism is a biholomorphic map (also called a conformal map), from a surface to itself. For example...

a Riemann surface, every point admits an open neighborhood which is biholomorphic to an open subset of the complex plane. Thereby the notion of a meromorphic...

4064/sm-51-3-241-250 Fefferman, Charles (1974), "The Bergman kernel and biholomorphic mappings of pseudoconvex domains", Inventiones Mathematicae, 26 (1):...

of G {\displaystyle G} as a neighbourhood of 0 {\displaystyle 0} ). Biholomorphic mapping – Bijective holomorphic function with a holomorphic inversePages...

theorem says that a simply connected domain in the complex plane is "biholomorphically equivalent" (i.e. there is a bijection between them that is holomorphic...

half-plane H = {x + iy : y > 0} to itself, and is equal to the group of all biholomorphic (or equivalently: bijective, conformal and orientation-preserving) maps...

{\displaystyle X} is a complex manifold and if the local trivializations are biholomorphic. A complex vector bundle can be thought of as a real vector bundle with...

is, if c belongs to the (usual) Mandelbrot set), then there exist a biholomorphic map ψ between the outer Fatou domain and the outer of the unit circle...

functions cannot have certain kinds of well-behaved invariant manifolds. A biholomorphic mapping F : C n → C n {\displaystyle F:\mathbb {C} ^{n}\to \mathbb {C}...

functions on Rn. A complex manifold is an atlas compatible with the biholomorphic functions on open sets in Cn. And so forth. Thus, pseudogroups provide...

considered as a Riemann surface, the open unit disk is isomorphic ("biholomorphic", or "conformally equivalent") to the upper half-plane, and the two...

Hermitian symmetric space X {\displaystyle X} of noncompact type is biholomorphic to a bounded domain in some C n {\displaystyle \mathbb {C} ^{n}} , and...

biholomorphically equivalent or that they are biholomorphic. When n > 1 {\displaystyle n>1} , open balls and open polydiscs are not biholomorphically...

finding a Siegel domain of type 2 in 4 dimensions that is homogeneous and biholomorphic to a bounded domain but not symmetric. In dimensions at least 7 there...

Włodzimierz (2008). "An Example of a Bounded C-Convex Domain Which is Not Biholomorphic to a Convex Domain". Mathematica Scandinavica. 102 (1): 149–155. doi:10...

(z,\theta +2\pi )} . As a complex manifold, R {\displaystyle R} is biholomorphic with C {\displaystyle \mathbb {C} } via log R {\displaystyle \log _{R}}...

for holomorphic 1-forms on complex manifolds — manifolds over C with biholomorphic transition functions. Specifically, if ω 1 , … , ω r {\displaystyle...

|f'(0)|=1} Bloch's theorem states that there is a disk S ⊂ D on which f is biholomorphic and f(S) contains a disk with radius 1/72. If f is a holomorphic function...

be embedded into C 2 n + 1 {\displaystyle \mathbb {C} ^{2n+1}} by a biholomorphic proper map. These facts imply that a Stein manifold is a closed complex...

bounded domain. More importantly, the distance dG is invariant under biholomorphic mappings of G to another domain G ′ {\displaystyle G'} . That is if...

Ewa (1980), "A simplification and extension of Fefferman's theorem on biholomorphic mappings", Inventiones Mathematicae, 57 (3): 283–289, doi:10.1007/BF01418930...

Mathematics. Vol. 122, No. 4 (16 pages). 5. A.V. Abrosimov (1988). On Locally Biholomorphic Equivalence of Smooth Hypersurfaces in С2. Proceedings of the USSR Academy...

the Poincaré half-plane model for the hyperbolic plane the group of biholomorphic transformations is the group P S L 2 ( R ) {\displaystyle \mathrm {PSL}...

group coincides with the special linear group SL2(R). This group biholomorphically acts on the complex upper half-plane by fractional-linear transformations...

{\displaystyle \phi _{U}:\pi ^{-1}(U)\to U\times \mathbf {C} ^{k}} are biholomorphic maps. This is equivalent to requiring that the transition functions...

this space is endowed with the structure of a complex manifold. It is biholomorphic to the unit ball of C n {\displaystyle \mathbb {C} ^{n}} , because every...

varieties of dimension n {\displaystyle n} containing an open dense subset biholomorphic to ( C ∗ ) n {\displaystyle (\mathbb {C} ^{*})^{n}} , equipped with...

holomorphically isomorphic (one also says: "conformally equivalent" or "biholomorphic") to one of the following: the Riemann sphere the complex plane the...

+ 1 ) α , {\displaystyle c_{1}(M)\geq (n+1)\alpha ,} then M must be biholomorphic to complex projective space. This, in combination with the Goldberg–Kobayashi...

of (N, h) is appropriately negative, then M and N must be biholomorphic or anti-biholomorphic if they are homotopic to each other; the biholomorphism (or...

manifold M {\displaystyle M} with charts U i {\displaystyle U_{i}} and biholomorphic maps f j k {\displaystyle f_{jk}} sending z k → z j = ( z j 1 , … ,...

Thus if X is the Riemann sphere, then X must be simply connected and biholomorphic to the Riemann sphere, the genus zero Riemann surface. When X is the...

P 2 ( C ) {\displaystyle \mathbb {P} ^{2}(\mathbb {C} )} but is not biholomorphic to it; the first example was discovered by Mumford. By work of Klingler...

was a complex surface homeomorphic to the projective plane but not biholomorphic to it. Yau (1977) showed that there was no such surface, so the closest...

(X,f)} that lead to combinatorially equivalent dessins d'enfants are biholomorphic, and Belyi's theorem implies that, for any compact Riemann surface X...

} Via a Cayley transform, H g {\displaystyle {\mathcal {H}}_{g}} is biholomorphic to a bounded symmetric domain, namely the Siegel disk { W ∈ M g ( C...

{\displaystyle \exp {\mathfrak {m}}_{+}} , a dense open subset of H / K biholomorphic to m + {\displaystyle {\mathfrak {m}}_{+}} . The corresponding domain...

group of biholomorphic transformations of D. The symmetry z ↦ −z is a biholomorphic Möbius transformation fixing only 0. The biholomorphic mappings of...

derivative and the other 1-cocycle defined on Diff(S1) can be extended to biholomorphic between open sets in the complex plane. In this case the local description...

proper subregions of the complex space of dimension n, which are biholomorphically equivalent to the whole space. (Such regions cannot exist for n=1...

Banach spaces." He proved important results in this area such as the biholomorphic embedding theorem for a Stein manifold as a closed submanifold in C...

moduli spaces. Namely, two polystable holomorphic vector bundles are biholomorphic if and only if there exists a gauge transformation taking the corresponding...

3-sphere in the surface with connected complement, with a neighbourhood biholomorphic to a neighbourhood of a sphere in C2. The global spherical shell conjecture...

index. A bounded symmetric domain is said to be of tube type if it is biholomorphically equivalent to a domain of the form T Ω = V + i Ω , {\displaystyle...